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If the volume of a Box is \(1,463,000\text{ cubic millimetres}\), what is the volume of the box in cubic meters? (\(1\text{ millimeter} = 0.001\text{ meter}\))
Let's start by understanding what we're being asked to do. We have a box with a volume of 1,463,000 cubic millimeters, and we need to find its volume in cubic meters.
We're given that \(1 \text{ millimeter} = 0.001 \text{ meter}\). This is our conversion factor for length.
The key insight here is that we're dealing with volume, not just length. Volume involves three dimensions - length, width, and height - so our conversion will be more complex than a simple linear conversion.
Process Skill: TRANSLATE - Converting the problem language into mathematical understanding
Here's where we need to think carefully. When we convert from millimeters to meters for length, we multiply by 0.001. But volume is three-dimensional!
Think of it this way: imagine a cube that's 1 meter on each side. Its volume is 1 cubic meter.
Now, let's measure that same cube in millimeters:
• Each side is 1,000 millimeters long
• So the volume is \(1{,}000 \times 1{,}000 \times 1{,}000 = 1{,}000{,}000{,}000\) cubic millimeters
This means: \(1 \text{ cubic meter} = 1{,}000{,}000{,}000 \text{ cubic millimeters}\)
We can verify this using our conversion factor: since \(1 \text{ mm} = 0.001 \text{ m}\), then \(1 \text{ m} = 1{,}000 \text{ mm}\)
So \(1 \text{ cubic meter} = (1{,}000)^3 \text{ cubic millimeters} = 1{,}000{,}000{,}000 \text{ cubic millimeters}\)
Process Skill: INFER - Drawing the non-obvious conclusion that volume conversion requires cubing the linear factor
From our understanding above, we know that:
\(1 \text{ cubic meter} = 1{,}000{,}000{,}000 \text{ cubic millimeters}\)
We can also express this using our given conversion factor:
• \(1 \text{ mm} = 0.001 \text{ m}\)
• Therefore: \(1 \text{ m} = 1{,}000 \text{ mm}\)
• So: \(1 \text{ m}^3 = (1{,}000 \text{ mm})^3 = 1{,}000^3 \text{ mm}^3 = 1{,}000{,}000{,}000 \text{ mm}^3\)
This confirms our conversion factor: 1 cubic meter = 1,000,000,000 cubic millimeters
Now we can convert our volume from cubic millimeters to cubic meters.
Given volume = 1,463,000 cubic millimeters
To convert to cubic meters, we divide by our conversion factor:
Volume in cubic meters = \(1{,}463{,}000 \div 1{,}000{,}000{,}000\)
Let's simplify this step by step:
\(= 1{,}463{,}000 \div (10^9)\)
\(= 1.463 \times 10^6 \div 10^9\)
\(= 1.463 \times 10^{6-9}\)
\(= 1.463 \times 10^{-3}\)
\(= 0.001463\)
The volume of the box is 0.001463 cubic meters.
Looking at our answer choices, this matches option E) 0.001463.
We can verify this makes sense: since millimeters are much smaller than meters, we expect many more cubic millimeters than cubic meters for the same volume, so our answer should be much smaller than 1,463,000 - and 0.001463 is indeed much smaller.
1. Linear vs. Cubic Conversion Confusion
Students often fail to recognize that volume conversion requires cubing the linear conversion factor. They might think: "If \(1 \text{ mm} = 0.001 \text{ m}\), then \(1 \text{ cubic mm} = 0.001 \text{ cubic m}\)" and simply multiply 1,463,000 by 0.001 to get 1,463 (answer choice B). This is the most common error because students don't internalize that volume involves three dimensions.
2. Misunderstanding the Conversion Direction
Some students get confused about whether to multiply or divide. Since they're converting from a smaller unit (mm³) to a larger unit (m³), they should expect a smaller numerical value. However, they might incorrectly think they need to multiply by 1,000,000,000 instead of dividing, leading to an astronomically large answer.
3. Incorrect Conversion Factor Calculation
Students might calculate the cubic conversion factor incorrectly. Instead of recognizing that \(1 \text{ m} = 1{,}000 \text{ mm}\), so \(1 \text{ m}^3 = (1{,}000)^3 = 1{,}000{,}000{,}000 \text{ mm}^3\), they might use \((0.001)^3 = 0.000000001\) as their conversion factor and multiply instead of using the reciprocal relationship.
1. Powers of 10 Arithmetic Errors
When dividing 1,463,000 by 1,000,000,000, students often make errors with the decimal placement. They might write 1,463,000 as \(1.463 \times 10^6\) correctly, but then struggle with \(10^6 \div 10^9 = 10^{-3}\), either forgetting the negative sign or miscounting the zeros, leading to answers like 0.1463 (choice C) or 1.463 (choice B).
2. Scientific Notation Conversion Mistakes
Students may correctly identify that they need \(1.463 \times 10^{-3}\) but then incorrectly convert this to decimal form. They might place the decimal point in the wrong position, getting 0.01463 (choice D) instead of 0.001463, because they move the decimal 2 places instead of 3 places to the left.
3. Calculation Setup Errors
Some students set up the division incorrectly by not properly organizing the numbers. They might calculate \(1{,}463{,}000 \div 1{,}000{,}000 = 1.463\) instead of \(1{,}463{,}000 \div 1{,}000{,}000{,}000 = 0.001463\), essentially using the wrong conversion factor due to miscounting zeros.
1. Reasonableness Check Failure
Students might arrive at the correct calculation but select the wrong answer choice because they don't perform a sanity check. For instance, they might select choice B (1.463) without realizing that this would mean the volume barely changed despite converting from much smaller units (mm³) to much larger units (m³), which should result in a much smaller number.
2. Decimal Reading Errors
When looking at 0.001463, students might misread the number of zeros after the decimal point and select 0.01463 (choice D) instead. This is especially common when students are rushing or not carefully counting the decimal places in both their calculated answer and the answer choices.